We prove that the -invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.
Nous montrons que l’invariant modulaire d’une courbe elliptique définie sur un corps quadratique imaginaire ayant par-tout bonne réduction vérifie certaines équations diophantiennes, sous réserve que soient vérifiées certaines hypothèses relatives à l’arithmétique du corps. En résolvant explicitement ces équations dans l’anneau des entiers du corps, nous montrons que de telles courbes n’existent pas sur certains corps quadratiques imaginaires. Nos résultats généralisent des résultats antérieurs de Setzer et Stroeker.
@article{JTNB_2001__13_1_201_0, author = {Masanari Kida}, title = {Good reduction of elliptic curves over imaginary quadratic fields}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {201--209}, publisher = {Universit\'e Bordeaux I}, volume = {13}, number = {1}, year = {2001}, zbl = {02081359}, mrnumber = {1838081}, language = {en}, url = {https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_201_0/} }
TY - JOUR AU - Masanari Kida TI - Good reduction of elliptic curves over imaginary quadratic fields JO - Journal de théorie des nombres de Bordeaux PY - 2001 SP - 201 EP - 209 VL - 13 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_201_0/ LA - en ID - JTNB_2001__13_1_201_0 ER -
%0 Journal Article %A Masanari Kida %T Good reduction of elliptic curves over imaginary quadratic fields %J Journal de théorie des nombres de Bordeaux %D 2001 %P 201-209 %V 13 %N 1 %I Université Bordeaux I %U https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_201_0/ %G en %F JTNB_2001__13_1_201_0
Masanari Kida. Good reduction of elliptic curves over imaginary quadratic fields. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 201-209. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_201_0/
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